Fast diffusion equations: matching large time asymptotics by relative entropy methods

Jean Dolbeault (1) , Giuseppe Toscani (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics,

A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

Mots clés: Fast diffusion equation; porous media equation; Barenblatt solutions; Hardy-Poincaré inequalities; large time behaviour; second moment; asymptotic expansion; intermediate asymptotics; sharp rates; optimal constants

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: matching large time asymptotics by relative entropy methods. Kinetic and Related Models , 2011, 4 (3), pp.701-716. ⟨hal-00482898v2⟩